Unless the terminals are corroded, I don't think the socket is the source of the problem.
It sounds like you just checked the voltage at/between the leads at the plug.
Connect the ground lead to chassis/ground, and the other test lead with the orange wire. Check the voltage you have coming off the orange wire. It should be 12V (at ALL times; regardless of headlight/high beam switch position) because this wire is the +12V supply. If you have +12V here than the problem is with the other wire (pink/black) which is the ground supply. It is an indirect ground connection that is switched at the high beam switch so it may be difficult to test. Because it is an indirect connection, it may not test zero resistance (which would be normal) if you check the resistance between this lead and chassis/ground.
Based on the wiring diagram, it looks like a bad fog lamp relay could be a possibility as well. You can temporarily swap in the horn relay on any other relay that is similar to see if that fixes the problem.
I've attached a pic of the wiring diagram. It's still legible, but a bit blurry though b/c I had to compress it alot to get under the 100 KB file size limit.
Good luck
Intrigue headlamp wiring1.jpg